The main purpose of this paper is to better understand groups that do not have the unique product property. In particular, the goal is to better understand Promislow's example, G, of such a group. In doing so, we will develop methods for generating examples of other sets that do not have the unique product property. With these methods we can show that there exists other distinct 14 element, square, non-unique product sets in G that are not inversions or translations. Also, this paper answers the question as to whether every non-unique product set can have only 14 elements in the negative by producing a 17 element square n.u.p. set. The secondary purpose of this paper is to demonstrate that in the group ring K[G], there are no units of support size 3.